This tag is for questions relating to "Parabolic partial differential equation", are usually time dependent and represent diffusion-like processes. Solutions are smooth in space but may possess singularities. However, information travels at infinite speed in a parabolic system.
By analogy with the conic sections (ellipse, parabola and hyperbola) partial differential equations have been classified as elliptic, parabolic and hyperbolic. Mathematically, parabolic PDEs serve as a transition from the hyperbolic PDEs to the elliptic PDEs. Physically, parabolic PDEs tend to arise in time dependent diffusion problems, such as the transient flow of heat in accordance with Fourier's law of heat conduction.
Suppose that $~u = u(x, t)~$ satisfies the second order partial differential equation $$Au_{xx} + Bu_{xt} + Cu_{tt} + Du_x + Eu_t + F u = G$$ in which $~A, ~\cdots ,~ G~$ are given functions.
This equation is said to be parabolic if $~B^2 − 4AC = 0~$
The archetypal parabolic evolution equation is the “heat conduction” or “diffusion” equation: $$\frac{∂u}{∂t}=\frac{∂^2u}{∂x^2}\qquad \text{($1$-dimensional)}~,$$ or more generally, for $~κ > 0~$, $$\frac{∂u}{∂t}= ∇ · (κ ∇u)= κ ∇^2u \quad\text{($κ$ constant)}~,$$ In one dimension, $$\frac{∂u}{∂t}=κ~\frac{∂^2u}{∂x^2}~$$
Problems which are well-posed for the heat equation will be well-posed for more general parabolic equation.
Reference:
https://en.wikipedia.org/wiki/Parabolic_partial_differential_equation
